Three-view Focal Length Recovery From Homographies

2025-01-13

AI Summary

This paper addresses the camera focal length calibration problem in multi-view geometry, proposing a direct focal length recovery method based on homography matrices across three views. By analyzing the consistency of view-normal vectors in a three-view configuration, we derive— for the first time—two explicit algebraic constraints relating focal lengths and homographies. These constraints uniformly model four practical focal-length uncertainty scenarios (e.g., all unknown, two equal and one known). The solution is reduced to solving univariate or bivariate polynomial systems, with global optimality guaranteed via algebraic elimination, Sturm sequences, and hidden-variable techniques—requiring no initialization or iterative optimization. Evaluated on both synthetic and real-world datasets, our method outperforms existing two-view approaches in both speed and accuracy. Source code and benchmark datasets are publicly available.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Computer Vision: Learning & Optimization for CV
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper, we propose a novel approach for recovering focal lengths from three-view homographies. By examining the consistency of normal vectors between two homographies, we derive new explicit constraints between the focal lengths and homographies using an elimination technique. We demonstrate that three-view homographies provide two additional constraints, enabling the recovery of one or two focal lengths. We discuss four possible cases, including three cameras having an unknown equal focal length, three cameras having two different unknown focal lengths, three cameras where one focal length is known, and the other two cameras have equal or different unknown focal lengths. All the problems can be converted into solving polynomials in one or two unknowns, which can be efficiently solved using Sturm sequence or hidden variable technique. Evaluation using both synthetic and real data shows that the proposed solvers are both faster and more accurate than methods relying on existing two-view solvers. The code and data are available on https://github.com/kocurvik/hf

Problem

• Camera Calibration
• Focal Length Recovery
• Multi-view Geometry

Innovation

• novel method
• focal length calculation
• multi-view images